Path integral representation and quantum-classical correspondence for nonadiabatic systems 1 Mikiya Fujii, Yamashita-Ushiyama Lab, Dept. of Chemical System Engineering, The Unviersity of Tokyo 1. Introduction to nonadiabatic transitions 2. Nonadiabatic path integral based on overlap integrals 3. Nonadiabatic partition functions: nonadiabatic beads model 4. Semiclassical nonadiabatic kernel: “rigorous” surface hopping 5. Semiclassical quantization of nonadiabatic systems: quantum-classical correspondence in nonadiabatic steady states
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Path integral representation and quantum-classical correspondence for nonadiabatic systems
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Mikiya Fujii, Yamashita-Ushiyama Lab, Dept. of Chemical System Engineering, The Unviersity of Tokyo
1. Introduction to nonadiabatic transitions
2. Nonadiabatic path integral based on overlap integrals
3. Nonadiabatic partition functions: nonadiabatic beads model
5. Semiclassical quantization of nonadiabatic systems: quantum-classical correspondence in nonadiabatic steady states
Transitions of nuclear wavepackets between electronic eigenstates (adiabatic surfaces)
Femtosecond time-resolved spectroscopy of the dynamics at conical intersections, G. Stock and W. Domcke, in: Conical Intersections, eds: W. Domcke, D. R. Yarkony, and H. Koppel, (World Scientific, Singapore, 2003) , figure from http://www.moldyn.uni-freiburg.de/research/ultrafast_nonadiabatic_photoreactions.html
5. Semiclassical quantization of nonadiabatic systems: quantum-classical correspondence in nonadiabatic steady states
CONTENTS
Semiclassical Quantization
Revealing correspondence between time-invariant structures in classical mechanics and steady states in quantum mechanics
e.g. Bohr’s model for Hydrogen, Bohr-Sommerfeld, Einstein–Brillouin–Keller, etc
H| i = E| i
steady states in quantum mechanics
q
p
time-invariant structures in phase space of
classical mechanics
periodic orbits torus
big← →small~
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Objective
Finding a quantum-classical correspondence for nonadiabatic steady states i.e. How time-invariant structures in nuclear phase space should be quantized
Especially, the semiclassical concepts of the nonadiabatic transition (i.e. classical dynamics on adiabatic surfaces and hopping) should be held. !The reason is that some pioneering studies that treat electrons and nuclei in equal-footing-manner have been already presented for the semiclassical quantization. e.g. Meyer-Miller (JCP 70, 3214 (1979)) and Stock-Thoss (PRL. 78, 578 (1997))
big← →small~nonadiabatic eigenstates
q
p
?nuclear phase space
Gutzwiller’s trace formulaSemiclassical approximation to DOS, which has revealed correspondence between quantum energy levels and classical periodic orbits through divergences of DOS.
classical action: Phase space volume
⌫ = 2
Scl = 2⇡E/!
e.g. Harmonic oscillator
Maslov index: number of intersects between trajectory and R-axis
geometric quantity of a cycle of primitive
periodic orbit
number of cycle of primitive periodic orbit
Sum of k-cycle diverges at quantum energy levels
1 = exp
✓i
~2⇡E
!� i⇡
◆�
) En =
✓n+
1
2
◆~!
}⌦(E) /
1X
k=0
exp
✓i
~Scl � i⇡
2
⌫
◆�k=
1� exp
✓i
~Scl � i⇡
2
⌫
◆��1
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⌦(E) /X
�2PHPO
1� ⇠� exp
✓i
~Scl� � i⇡
2
⌫�
◆��1
①Sum of “Primitive Hopping Periodic Orbits (PHPO)”
Taking the summation of geometric series related to k, naively, leads to
⇠� < 1This term does not diverge because .
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②Infinite product of the overlap integrals: ⇠ ⌘ lim�!1
�Y
k=0
h n(tk+1);R(tk+1)| n(tk);R(tk)i
There are 2 differences from the Gutzwiller’s (adiabatic) trace formula
⌦(E) /X
�2PHPO
1X
k=0
⇠� exp
✓i
~Scl� � i⇡
2
⌫�
◆�kNonadiabatic Trace formula
That is, individual PHPO cannot be quantized.
We must introduce another way to take the summation of infinite number of the PHPOs
Bit sequence which represents PHPO A concrete example: Two adiabatic harmonic oscillators which interact nonadiabatically at the origin only.D12 = �(R) sin(✓)
Ri Rj Rk Rl 0, 1, 1, and 0 are assigned when a trajectory passes through Ri, Rj, Rk, and Rl,
Periodic bit sequences representing PHPOs can be expressed with dots on the fist and last bits
0111 ⌘ 011101110111 · · ·01 ⌘ 0101010101 · · ·
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We can also confirm that the periodic and non-periodic orbits correspond to rational and irrational numbers, respectively, because periodic bit sequences correspond to rational number in binary digits. So, the number of periodic orbits is countable infinite while the number of arbitrary orbits is uncountable infinite.
D12 = �(R) sin(✓)
Ri Rj Rk Rl 0100011100110 in odd-numbered bits means “returning to Ri”.
Decomposition of each PHPO
01 + 00 + 0111 + 0011
At the 0 in odd-numbered bits, we can decompose this PHPO to “more primitive (prime) bits (PHPOs)”.
Threfore, arbitrary hopping periodic orbits passing through Ri can be represented by combinations of these prime PHPOs:
00, 01, 0110, 0111, 0010, 0011,where 1 means combinations of 11 and 10.
Hereafter, this set of prime PHOPs are represented as
S0 ⌘
(Ⅰ) All prime PHPOs in ”Si” pass through the same phase space point (Ⅱ) Any pair of prime PHPOs (Γ, Γ’) in ”Si” is coprime: